On the average distance property and the size of the unit sphere.

For every finite subset F of the unit sphere S of a Banach space and for every x 2 S consider the average distance \u3bc(F, x) between the points of F and x. The set of average numbers for a Banach space X is the closed interval [\u3bc1(X), \u3bc2(X)], where \u3bc1(X) = supF infx \u3bc(F, x) and \u3bc2(X) = infF supx \u3bc(F, x) as F runs over all finite subsets of S and x runs over S. First we prove some general results about the average numbers, including the fact that \u3bc1 and \u3bc2 depend continuously on the Banach-Mazur distance between spaces. Then we compute \u3bc1(X) and \u3bc2(X) and other related parameters for some Banach spaces X such as c0, l1, l1, L[0, 1] and C[0, 1]. We also discuss how the average numbers are related to the modulus of convexity, to orthogonality, and to the existence of extreme points. Finally, we prove that for every Banach space X the average numbers of the bidual X are also average numbers of X